Stochastic () refers to the property of being well described by a

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* ''Formalized Music: Thought and Mathematics in Composition'' by Iannis Xenakis, * ''Frequency and the Emergence of Linguistic Structure'' by Joan Bybee and Paul Hopper (eds.), / (Eur.) * The Stochastic Empirical Loading and Dilution Model provides documentation and computer code for modeling stochastic processes in Visual Basic for Applications.

random
In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. I ...

probability distribution
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselves, these two terms are often used synonymously. Furthermore, in probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of ...

, the formal concept of a ''stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a Indexed family, family of random variables. Stochastic processes are widely used as mathematical models of systems and pheno ...

'' is also referred to as a ''random process''.
Stochasticity is used in many different fields, including the natural sciences
Natural science is a Branches of science, branch of science concerned with the description, understanding and prediction of Phenomenon, natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer r ...

such as biology
Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, Developmenta ...

, chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds composed of atoms, molecules and i ...

, ecology
Ecology (from el, οἶκος, "house" and el, -λογία, label=none, "study of") is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms
In biology
...

, neuroscience
Neuroscience is the science, scientific study of the nervous system. It is a Multidisciplinary approach, multidisciplinary science that combines physiology, anatomy, molecular biology, developmental biology, cytology, computer science and Mathem ...

, and physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scien ...

, as well as technology
Technology ("science of craft", from Ancient Greek, Greek , ''techne'', "art, skill, cunning of hand"; and , ''wikt:-logia, -logia'') is the sum of any Art techniques and materials, techniques, skills, Scientific method, methods, and Business ...

and engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...

fields such as image processing
Digital image processing is the use of a digital computer
A computer is a machine
A machine is a man-made device that uses power to apply forces and control movement to perform an action. Machines can be driven by animals and people
...

, signal processing
Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniques c ...

, information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamentally established by the ...

, computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of Algorithm, algorithmic proc ...

, cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia
''-logy'' is a suffix
In linguistics
Linguistics is the science, scientific study of language. It encompa ...

, and telecommunication
Telecommunication is the transmission of information by various types of technologies over wire
A wire is a single usually cylindrical, flexible strand or rod of metal. Wires are used to bear mechanical loads or electricity
Electr ...

s. It is also used in finance, due to seemingly random changes in financial market
A financial market is a market
Market may refer to:
*Market (economics)
*Market economy
*Marketplace, a physical marketplace or public market
Geography
*Märket, an island shared by Finland and Sweden
Art, entertainment, and media Films
*Ma ...

s as well as in medicine, linguistics, music, media, colour theory, botany, manufacturing, and geomorphology. Stochastic modeling is also used in social science
Social science is the Branches of science, branch of science devoted to the study of society, societies and the Social relation, relationships among individuals within those societies. The term was formerly used to refer to the field of sociol ...

.
Etymology

The word ''stochastic'' in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek word meaning "to aim at a mark, guess", and the Oxford English Dictionary gives the year 1662 as its earliest occurrence. In his work on probability ''Ars Conjectandi'', originally published in Latin in 1713,Jakob Bernoulli
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includ ...

used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics". This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz
Ladislaus Josephovich Bortkiewicz (7 August 1868 – 15 July 1931) was a Russian economist
An economist is a practitioner in the social sciences, social science discipline of economics.
The individual may also study, develop, and apply the ...

, who in 1917 wrote in German the word ''stochastik'' with a sense meaning random. The term ''stochastic process'' first appeared in English in a 1934 paper by Joseph Doob
Joseph Leo "Joe" Doob (February 27, 1910 – June 7, 2004) was an American
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the United States
The United States of Ame ...

. For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term ''stochastischer Prozeß'' was used in German by Aleksandr Khinchin
Aleksandr Yakovlevich Khinchin (russian: Алекса́ндр Я́ковлевич Хи́нчин, french: Alexandre Khintchine; July 19, 1894 – November 18, 1959) was a Soviet mathematician
A mathematician is someone who uses an extensive ...

, though the German term had been used earlier in 1931 by Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovie ...

.
Mathematics

In the early 1930s, Aleksandr Khinchin gave the first mathematical definition of a stochastic process as a family of random variables indexed by the real line. Further fundamental work on probability theory and stochastic processes was done by Khinchin as well as other mathematicians such asAndrey Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovie ...

, Joseph Doob
Joseph Leo "Joe" Doob (February 27, 1910 – June 7, 2004) was an American
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the United States
The United States of Ame ...

, William Feller
William "Vilim" Feller (July 7, 1906 – January 14, 1970), born Vilibald Srećko Feller, was a Croatia
:* french: link=no, République de Croatie
:* hu, Horvát Köztársaság
:* it, Repubblica di Croazia
:* rue, Републіка Х ...

, Maurice FréchetMaurice may refer to:
People
*Saint Maurice
Saint Maurice (also Moritz, Morris, or Mauritius; ) was the leader of the legendary Roman Theban Legion in the 3rd century, and one of the favorite and most widely venerated saints of that group. He w ...

, Paul Lévy, Wolfgang Doeblin
Wolfgang Doeblin, known in France as Vincent Doblin (17 March 1915 – 21 June 1940), was a French people, French-Germans, German mathematician.
Life
A native of Berlin, Wolfgang was the son of the Jewish-German novelist and physician, Alfred D ...

, and Harald Cramér
Harald Cramér (; 25 September 1893 – 5 October 1985) was a Sweden, Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. John Kingman described him as "one of the giants of st ...

. Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory".
In mathematics, the theory of stochastic processes is an important contribution to probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of ...

, and continues to be an active topic of research for both theory and applications.
The word ''stochastic'' is used to describe other terms and objects in mathematics. Examples include a stochastic matrix
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, which describes a stochastic process known as a Markov process
A Markov chain or Markov process is a stochastic model
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory tre ...

, and stochastic calculus, which involves differential equation
In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ...

s and integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s based on stochastic processes such as the Wiener process
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, also called the Brownian motion process.
Natural science

One of the simplest continuous-time stochastic processes isBrownian motion
File:Brownian motion large.gif, This is a simulation of the Brownian motion of a big particle (dust particle) that collides with a large set of smaller particles (molecules of a gas) which move with different velocities in different random direc ...

. This was first observed by botanist Robert Brown while looking through a microscope at pollen grains in water.
Physics

TheMonte Carlo method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computation
Computation is any type of calculation
A calculation is a deliberate process that transforms one or more inputs into one or more results. The term is used in a v ...

is a stochastic method popularized by physics researchers Stanisław Ulam, Enrico Fermi
Enrico Fermi (; 29 September 1901 - 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and ...

, John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian Americans, Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. Von Neumann was generally rega ...

, and Nicholas Metropolis
Nicholas Constantine Metropolis ( Greek: ; June 11, 1915 – October 17, 1999) was a Greek-American physicist
A physicist is a scientist
A scientist is a person who conducts Scientific method, scientific research to advance knowledge in ...

. The use of randomness
In common parlance, randomness is the apparent or actual lack of pattern
A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pat ...

and the repetitive nature of the process are analogous to the activities conducted at a casino.
Methods of simulation and statistical sampling generally did the opposite: using simulation to test a previously understood deterministic problem. Though examples of an "inverted" approach do exist historically, they were not considered a general method until the popularity of the Monte Carlo method spread.
Perhaps the most famous early use was by Enrico Fermi in 1930, when he used a random method to calculate the properties of the newly discovered neutron
The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behav ...

. Monte Carlo methods were central to the simulation
A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or process, wh ...

s required for the Manhattan Project
The Manhattan Project was a research and development
Research is " creative and systematic work undertaken to increase the stock of knowledge". It involves the collection, organization, and analysis of information to increase understa ...

, though they were severely limited by the computational tools of the time. Therefore, it was only after electronic computers were first built (from 1945 on) that Monte Carlo methods began to be studied in depth. In the 1950s they were used at Los Alamos for early work relating to the development of the hydrogen bomb
lenses2) Uranium-238 ("tamper") lined with beryllium reflector3) Vacuum ("levitated core")4) Tritium "boost" gas (blue) within plutonium or uranium hollow core 5) Radiation channel filled with polystyrene foam6) Uranium ("pusher/tamper")7) Lithium ...

, and became popularized in the fields of physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scien ...

, physical chemistry
Physical chemistry is the study of macroscopic
The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopi ...

, and operations research. The RAND Corporation
The RAND Corporation ("research and development") is an American nonprofit
A nonprofit organization (NPO), also known as a non-business entity, not-for-profit organization, or nonprofit institution, is a legal entity organized and ope ...

and the U.S. Air Force
The United States Air Force (USAF) is the air
File:Atmosphere gas proportions.svg, Composition of Earth's atmosphere by volume, excluding water vapor. Lower pie represents trace gases that together compose about 0.043391% of the atmosphe ...

were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields.
Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the development of pseudorandom number generator
A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random number generation, random nu ...

s, which were far quicker to use than the tables of random numbers which had been previously used for statistical sampling.
Biology

Stochastic resonance: In biological systems, introducing stochastic "noise" has been found to help improve the signal strength of the internal feedback loops for balance and othervestibular
The Vestibular (from pt, vestíbulo, "entrance hall") is a competitive examination and is the primary and widespread entrance system used by Brazil
Brazil ( pt, Brasil; ), officially the Federative Republic of Brazil (Portuguese: ), is the ...

communication. It has been found to help diabetic and stroke patients with balance control. Many biochemical events also lend themselves to stochastic analysis. Gene expression
Gene expression is the process by which information from a gene
In biology, a gene (from ''genos'' "...Wilhelm Johannsen coined the word gene to describe the Mendelian_inheritance#History, Mendelian units of heredity..." (Greek language, ...

, for example, has a stochastic component through the molecular collisions—as during binding and unbinding of RNA polymerase
In molecular biology
Molecular biology is the branch of biology that seeks to understand the molecule, molecular basis of biological activity in and between Cell (biology), cells, including biomolecule, molecular synthesis, modification, mec ...

to a gene promoter
In genetics, a promoter is a sequence of DNA to which proteins bind that initiate transcription (genetics), transcription of a single RNA from the DNA downstream of it. This RNA may encode a protein, or can have a function in and of itself, such ...

—via the solution's Brownian motion
File:Brownian motion large.gif, This is a simulation of the Brownian motion of a big particle (dust particle) that collides with a large set of smaller particles (molecules of a gas) which move with different velocities in different random direc ...

.
Creativity

Simonton (2003, ''Psych Bulletin'') argues that creativity in science (of scientists) is a constrained stochastic behaviour such that new theories in all sciences are, at least in part, the product of astochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a Indexed family, family of random variables. Stochastic processes are widely used as mathematical models of systems and pheno ...

.
Computer science

Stochastic ray tracing is the application ofMonte Carlo simulation
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computation
Computation is any type of calculation
A calculation is a deliberate process that transforms one or more inputs into one or more results. The term is used in a v ...

to the computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great deal ...

ray tracing algorithm. "Distributed ray tracing
Distributed ray tracing, also called distribution ray tracing and stochastic ray tracing, is a refinement of Ray tracing (graphics), ray tracing that allows for the rendering (computer graphics), rendering of "soft" phenomena.
Conventional ray trac ...

samples the integrand
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

at many randomly chosen points and averages the results to obtain a better approximation. It is essentially an application of the Monte Carlo method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computation
Computation is any type of calculation
A calculation is a deliberate process that transforms one or more inputs into one or more results. The term is used in a v ...

to 3D computer graphics
3D computer graphics, sometimes called CGI, 3DCG or three-dimensional computer graphics (in contrast to 2D computer graphics
2D computer graphics is the Computer-generated imagery, computer-based generation of digital images—mostly from t ...

, and for this reason is also called ''Stochastic ray tracing''."
Stochastic forensicsStochastic forensics is a method to Forensics, forensically reconstruct digital activity lacking Digital artifact, artifacts, by analyzing Emergent properties#Emergent properties and processes, emergent properties resulting from the stochastic nature ...

analyzes computer crime by viewing computers as stochastic processes.
In artificial intelligence
Artificial intelligence (AI) is intelligence
Intelligence has been defined in many ways: the capacity for logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, ...

, stochastic programs work by using probabilistic methods to solve problems, as in simulated annealing
Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem ...

, stochastic neural networks, stochastic optimization
Stochastic optimization (SO) methods are optimization method
Method ( grc, μέθοδος, methodos) literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In recent centuries it more often means a presc ...

, genetic algorithm
spacecraft antenna. This complicated shape was found by an evolutionary computer design program to create the best radiation pattern. It is known as an evolved antenna.
In computer science
Computer science deals with the theoretical found ...

s, and genetic programming
In artificial intelligence, genetic programming (GP) is a technique of evolving programs, starting from a population of unfit (usually random) programs, fit for a particular task by applying operations analogous to natural genetic processes to the ...

. A problem itself may be stochastic as well, as in planning under uncertainty.
Finance

The financial markets use stochastic models to represent the seemingly random behaviour of assets such asstock
In finance, stock (also capital stock) consists of all of the shares
In financial markets
A financial market is a market in which people trade financial securities and derivatives at low transaction costs. Some of the securities in ...

s, commodities
In economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods ...

, relative currency
A currency, "in circulation", from la, currens, -entis, literally meaning "running" or "traversing" in the most specific sense is money
Image:National-Debt-Gillray.jpeg, In a 1786 James Gillray caricature, the plentiful money bags handed t ...

prices (i.e., the price of one currency compared to that of another, such as the price of US Dollar compared to that of the Euro), and interest rate
An interest rate is the amount of interest
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investm ...

s. These models are then used by quantitative analyst
Quantitative may refer to:
* Quantitative research, scientific investigation of quantitative properties
* Quantitative analysis (disambiguation)
* Meter (poetry), Quantitative verse, a metrical system in poetry
* Statistics, also known as quantita ...

s to value options on stock prices, bond prices, and on interest rates, see Markov models
In probability theory, a Markov model is a stochastic model used to Mathematical model, model pseudo-randomly changing systems. It is assumed that future states depend only on the current state, not on the events that occurred before it (that is, it ...

. Moreover, it is at the heart of the insurance industry
Insurance is a means of protection from financial loss. It is a form of risk management
Risk management is the identification, evaluation, and prioritization of risk
In simple terms, risk is the possibility of something bad happening. ...

.
Geomorphology

The formation of river meanders has been analyzed as a stochastic processLanguage and linguistics

Non-deterministic approaches in language studies are largely inspired by the work ofFerdinand de Saussure
Ferdinand de Saussure (; ; 26 November 1857 – 22 February 1913) was a Swiss
Swiss may refer to:
* the adjectival form of Switzerland
,german: Schweizer(in),french: Suisse(sse), it, svizzero/svizzera or , rm, Svizzer/Svizra
, government_typ ...

, for example, in functionalist linguistic theory, which argues that competence is based on Linguistic performance, performance. This distinction in functional theories of grammar should be carefully distinguished from the langue and parole, ''langue'' and ''parole'' distinction. To the extent that linguistic knowledge is constituted by experience with language, grammar is argued to be probabilistic and variable rather than fixed and absolute. This conception of grammar as probabilistic and variable follows from the idea that one's competence changes in accordance with one's experience with language. Though this conception has been contested, it has also provided the foundation for modern statistical natural language processing and for theories of language learning and change.
Manufacturing

Manufacturing processes are assumed to bestochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a Indexed family, family of random variables. Stochastic processes are widely used as mathematical models of systems and pheno ...

es. This assumption is largely valid for either continuous or batch manufacturing processes. Testing and monitoring of the process is recorded using a process control chart which plots a given process control parameter over time. Typically a dozen or many more parameters will be tracked simultaneously. Statistical models are used to define limit lines which define when corrective actions must be taken to bring the process back to its intended operational window.
This same approach is used in the service industry where parameters are replaced by processes related to service level agreements.
Media

The marketing and the changing movement of audience tastes and preferences, as well as the solicitation of and the scientific appeal of certain film and television debuts (i.e., their opening weekends, word-of-mouth, top-of-mind knowledge among surveyed groups, star name recognition and other elements of social media outreach and advertising), are determined in part by stochastic modeling. A recent attempt at repeat business analysis was done by Japanese scholars and is part of the Cinematic Contagion Systems patented by Geneva Media Holdings, and such modeling has been used in data collection from the time of the original Nielsen ratings to modern studio and television test audiences.Medicine

Stochastic effect, or "chance effect" is one classification of radiation effects that refers to the random, statistical nature of the damage. In contrast to the deterministic effect, severity is independent of dose. Only the ''probability'' of an effect increases with dose.Music

In music, mathematics, mathematical processes based on probability can generate stochastic elements. Stochastic processes may be used in music to compose a fixed piece or may be produced in performance. Stochastic music was pioneered by Iannis Xenakis, who coined the term ''stochastic music''. Specific examples of mathematics, statistics, and physics applied to music composition are the use of the statistical mechanics of gases in ''Pithoprakta'', statistical distribution of points on a plane in ''Diamorphoses'', minimal Constraint (mathematics), constraints in ''Achorripsis'', the normal distribution in ''ST/10'' and ''Atrées'', Markov chains in ''Analogiques'', game theory in ''Duel'' and ''Stratégie'', group theory in ''Nomos Alpha'' (for Siegfried Palm), set theory in ''Herma'' and ''Eonta'', andBrownian motion
File:Brownian motion large.gif, This is a simulation of the Brownian motion of a big particle (dust particle) that collides with a large set of smaller particles (molecules of a gas) which move with different velocities in different random direc ...

in ''N'Shima''. Xenakis frequently used computer music, computers to produce his scores, such as the ''ST'' series including ''Morsima-Amorsima'' and ''Atrées'', and founded CEMAMu. Earlier, John Cage and others had composed ''aleatoric music, aleatoric'' or indeterminate music, which is created by chance processes but does not have the strict mathematical basis (Cage's ''Music of Changes'', for example, uses a system of charts based on the ''I-Ching''). Lejaren Hiller and Leonard Issacson used generative grammars and Markov chains in their 1957 ''Illiac Suite''. Modern electronic music production techniques make these processes relatively simple to implement, and many hardware devices such as synthesizers and drum machines incorporate randomization features. Generative music techniques are therefore readily accessible to composers, performers, and producers.
Social sciences

Stochastic social science theory is similar to systems theory in that events are interactions of systems, although with a marked emphasis on unconscious processes. The event creates its own conditions of possibility, rendering it unpredictable if simply for the number of variables involved. Stochastic social science theory can be seen as an elaboration of a kind of 'third axis' in which to situate human behavior alongside the traditional 'nature vs. nurture' opposition. See Julia Kristeva on her usage of the 'semiotic', Luce Irigaray on reverse Heideggerian epistemology, and Pierre Bourdieu on polythetic space for examples of stochastic social science theory. The term "Stochastic Terrorism" has fallen into frequent use published August 12, 2019 CNN with regard to Lone wolf (terrorism), lone wolf terrorism. The terms "Scripted Violence" and "Stochastic Terrorism" are linked in a "cause <> effect" relationship. "Scripted Violence" rhetoric can result in an act of "Stochastic Terrorism." The phrase "scripted violence" has been used in social science since at least 2002. Author David Neiwert, who wrote the book ''Alt-America'', told Salon interviewer Chauncey Devega:Subtractive color reproduction

When color reproductions are made, the image is separated into its component colors by taking multiple photographs filtered for each color. One resultant film or plate represents each of the cyan, magenta, yellow, and black data. Color printing is a binary system, where ink is either present or not present, so all color separations to be printed must be translated into dots at some stage of the work-flow. Traditional line screens which are amplitude modulation, amplitude modulated had problems with moiré but were used until stochastic screening became available. A stochastic (or frequency modulation, frequency modulated) dot pattern creates a sharper image.See also

* Jump process * Sortition * Stochastic processNotes

References

Further reading

* . from Index Funds AdvisorIFA.com

* ''Formalized Music: Thought and Mathematics in Composition'' by Iannis Xenakis, * ''Frequency and the Emergence of Linguistic Structure'' by Joan Bybee and Paul Hopper (eds.), / (Eur.) * The Stochastic Empirical Loading and Dilution Model provides documentation and computer code for modeling stochastic processes in Visual Basic for Applications.

External links

* {{Authority control Stochastic processes, * Mathematical terminology